In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the director circle of an
ellipse or
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
(also called the
orthoptic circle or Fermat–Apollonius circle) is a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
consisting of all points where two
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
s to the ellipse or hyperbola cross each other.
Properties
The director circle of an ellipse
circumscribes the
minimum bounding box
In geometry, the minimum or smallest bounding or enclosing box for a point set in dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dimensions) within which all the points lie. When other kinds of measure ...
of the ellipse. It has the same center as the ellipse, with radius
, where
and
are the
semi-major axis and
semi-minor axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lo ...
of the ellipse. Additionally, it has the property that, when viewed from any point on the circle, the ellipse spans a
right angle.
The director circle of a hyperbola has radius , and so, may not exist in the
Euclidean plane, but could be a circle with imaginary radius in the
complex plane.
Generalization
More generally, for any collection of points , weights , and constant , one can define a circle as the locus of points such that
:
The director circle of an ellipse is a special case of this more general construction with two points and at the foci of the ellipse, weights , and equal to the square of the major axis of the ellipse. The
Apollonius circle, the locus of points such that the ratio of distances of to two foci and is a fixed constant , is another special case, with , , and .
Related constructions
In the case of a
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
the director circle degenerates to a straight line, the
directrix of the parabola.
Notes
References
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*{{citation, first=George Albert, last=Wentworth, title=Elements of Analytic Geometry, publisher=Ginn & Company, year=1886, page=150.
Conic sections
Circles